CIRCLE-VALUED MORSE THEORY, REIDEMEISTER TORSION, AND SEIBERG–WITTEN INVARIANTS OF 3-MANIFOLDS

Let X be a closed oriented Riemannian manifold with χ(X)=0 and b1(X)>0, and let φ : X→S1 be a circle-valued Morse function. Under some mild assumptions on φ, we prove a formula relating1. mber of closed orbits of the gradient flow of φ in different homology classes; rsion of the Novikov complex, which counts gradient flow lines between critical points of φ; and of Reidemeister torsion of X determined by the homotopy class of φ. dim(X)=3, we state a conjecture related to Taubes’s “SW=Gromov” theorem, and we use it to deduce (for closed manifolds, modulo signs) the Meng–Taubes relation between the Seiberg-Witten invariants and the “Milnor torsion” of X.